"The member has the best dog."

Translation: Das Mitglied hat den besten Hund.

[bmwx11]

Why wouldn't it be den bestenen? http://german.about.com/library/weekly/aa033098.htm

[myra]

Das Mitglied hat:
-den besten Hund (masculine)
-die beste Katze (feminine)
-das beste Pferd (neuter)
-die besten Hunde (plural)

As your link points out, the ending pattern is -en,-e,-e,-en. However these special kinds of adjectives (superlatives) differ from normal adjectives in that they cannot be used as predicates:

correct: Der Hund ist gut.
incorrect: Der Hund ist besten/best.
correct: Der Hund ist am besten.

[OsmanIlyas]

+1 thanks